Far-field approximations to the Kirchoff-Helmholtz representations of scattered fields

The standard far-field approximation to the Kirchhoff formula for the field scattered by a flat metallic plateSof arbitrary shape is given by a certain surface (double) integral. This double integral can be reduced to a line integral evaluated around the boundary of S. Moreover, ifSis a polygon, this line integral can be reduced to a closed form expression involving no integrations at all. The use of such line integral representations can easily reduce the costs of numerical calculation by orders of magnitude. If the integrands are to be sampledptimes per wavelength to achieve an acceptable degree of precision, and ifAis the area ofS, then the numerical evaluation of the double integral requiresp^{2}A/lambda^{2}functional evaluations whereas the line integral only requirespsqrt{A/lambda^{2}}. IfSis a polygon withNvertices, then only2Nfunctional evaluations are required to evaluate the closed form expression with no quadrature error at all.